Generating weakly triangulated graphs
نویسنده
چکیده
We show that a graph is weakly triangulated, or weakly chordal, if and only if it can be generated by starting with a graph with no edges, and repeatedly adding an edge, so that the new edge is not the middle edge of any chordless path with four vertices. This is a corollary of results due to Sritharan and Spinrad, and Hayward, Hoo ang and Maaray, and a natural analogue of a theorem due to Fulkerson and Gross, which states that a graph is triangulated, or chordal, if and only if it can be generated by starting with a graph with no vertices, and repeatedly adding a vertex, so that the new vertex is not the middle vertex of any chordless path with three vertices. Our result answers the question of whether there exists a composition scheme that generates exactly the class of weakly triangulated graphs. A graph is triangulated, or chordal, if it does not contain a chordless cycle with four or more vertices. The following well-known result of Fulkerson and Gross, via a theorem of Dirac, gives a way to build up triangulated graphs from the empty graph: Theorem 1. FG] D] A graph is triangulated if and only if it can be generated in the following manner: Start with a graph G 0 with no vertices. Repeatedly add a vertex v j to G j?1 to create the graph G j , such that v j is not the middle vertex of any P 3 of G j. A graph is weakly triangulated, or weakly chordal, if neither the graph nor its complement contains a chordless cycle with ve or more vertices. The class of weakly triangulated graphs, introduced in H], contains the class of triangulated graphs, and also the class of complements of triangulated graphs. Hayward, Hoo ang and Maaray showed: Theorem 2. HHM] Every weakly triangulated graph either is a clique or has a pair of vertices fv;wg, such that the removal of their common neighbors leaves them in diierent components.
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ورودعنوان ژورنال:
- Journal of Graph Theory
دوره 21 شماره
صفحات -
تاریخ انتشار 1996